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%I #9 Dec 19 2017 16:22:09
%S 1,2,4,-8,2448,130976,-2342848,-239130240,99052990720,8918588764672,
%T -2795242017684480,-92786315822417920,279479081010906828800,
%U -57316070780459900928,-39411396653183724314673152,5932051008707372732672475136,10689040617354387626585873252352
%N Expansion of e.g.f. arcsinh(x*sec(x)) (odd powers only).
%F a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x*sec(x)).
%e arcsinh(x*sec(x)) = x/1! + 2*x^3/3! + 4*x^5/5! - 8*x^7/7! + 2448*x^9/9! + ...
%t nmax = 17; Table[(CoefficientList[Series[ArcSinh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%o (PARI) first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asinh(x/cos(x)))), (4^n - 1)/3) \\ _Iain Fox_, Dec 19 2017
%Y Cf. A001818, A003700, A009118, A009119, A009562, A009563, A009765, A009843, A102072, A191003, A296464, A296466, A296679, A296680, A296741, A296743.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Dec 19 2017
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